1. Exercise 5

2. 5.1
The fuzzy relation R is denned on sets X1 = {a, b, c}, X2 = {s, t}, X3 = {x, y}, X4 = {i, j } as follows:
R(X1, X2, X3, X4) =
.4/ b, t, y, i +.6/ a, s, x, I + .9/ b, s, y, j + .6/ a, t, y, j + .2/ c, s, y, i.
(a) Compute the projections R1, 2, 4 , R1, 3 and R4.
(b) Compute the cylindric extensions [R1, 2, 4↑{X3}], [R1, 3↑{X2, X4}], [R4↑{ X1, X2, X3}].
(c) Compute the cylindric closure from the three cylindric extensions in (b).
(d) Is the cylindric closure from (c) equal to the original R?

3. 5.4
Consider matrices M1, M2, M3 in Table 5.4 as pages in a three-dimensional array that represents a fuzzy ternary relation. Determine:
(a) all two-dimensional projections;
(b) cylindric extensions and cylindric closure of the two-dimensional projections;
(c) all one-dimensional projections;
(d) cylindric extensions and cylindric closure of the one-dimensional projections;
(e) two three-dimensional arrays expressing the difference between each of the cylindric closures and the original ternary relation.

5. 5.6
The fuzzy binary relation R is defined on sets X ={1, 2 , , 100} and Y = {50, 5 1 , , 100} and represents the relation "x is much smaller than y" It is defined by membership function
(a) What is the domain of R ?
(b) What is the range of R ?
(c) What is the height of R ?
(d) Calculate R-1.

6. 5.11
Assuming X = Y, perform the max-min composition of some sequences of comparable relations given in Table 5.4. For example:
(a) M1。M2。M4。M5。M8。M9。M12。M13;
(b) M1。M2。M3。M4。M5。M6。M7,etc.

7. 5.19
The transitive closure of the relation denned by matrix M12 in Table 5.4 (Exercise 5.16) is a equivalence relation. Determine its partition tree.

8. 5.20
Relations defined by matrices M3 and M12 in Table 5.4 are compatibility relations (assume X = Y). Determine:
(a) simple diagrams of the relations;
(b) all complete α-covers of the relations.